Harmonic approximation and its applications
These lectures survey some recent developments concerning the theory and applications of harmonic approximation in Euclidean space. We begin with a discussion of the significance of the concept of thinness for harmonic approximation, and present a complete description of the closed (possibly unbounded) sets on which uniform harmonic approximation is possible. Next we demonstrate the power of such results by describing their use to solve an old problem concerning the Dirichlet problem for unbounded regions. The third lecture characterizes the functions on a given set which can be approximated by harmonic functions on a fixed open superset. Finally, we return to applications, and explain how some problems concerning the boundary behaviour of harmonic functions have recently been solved using harmonic approximation.
KeywordsHarmonic Function Dirichlet Problem Harmonic Measure Harmonic Approximation Boundary Behaviour
Unable to display preview. Download preview PDF.
- N. U. Arakelyan, Approximation complexe et propriétés des fonctions analytiques, Actes, Congrès intern. Math. (1970), Tome 2, 595–600.Google Scholar
- D. H. Armitage, Uniform and tangential harmonic approximation, These proceedings.Google Scholar
- T. Bagby and P. M. Gauthier, Harmonic approximation on closed subsets of Riemannian manifolds, in: Complex Potential Theory (P. M. Gauthier, ed.), NATO ASI Ser. C Math. Phys. Sci. 439, Kluwer, Dordrecht, 1994; 75–87.Google Scholar
- T. Bagby, P. M. Gauthier and J. Woodworth, Tangential harmonic approximation on Riemannian manifolds, in: Harmonic Analysis and Number Theory (S. W. Drury and M. Ram Murty, eds.), CMS Conf. Proc. 21, Amer. Math. Soc, Providence, RI, 1997; 58–72.Google Scholar
- B. Böe, Sets of determination for smooth harmonic functions, preprint.Google Scholar
- T. Carleman, Sur un théorème de Weierstrass, Ark. Mat. Astronom. Fys. 20B (1927), 1–5.Google Scholar
- S. J. Gardiner, Non-tangential limits, fine limits and the Dirichlet integral, to appear in Proc. Amer. Math. Soc. Google Scholar
- S. J. Gardiner and W. Hansen, Boundary sets where harmonic functions may become infinite, preprint.Google Scholar
- M. V. Keldys, On the solvability and stability of the Dirichlet problem, Uspekhi Mat. Nauk 8 (1941), 171–231 (Russian); English translation Amer. Math. Soc. Transl. 51 (1966), 1-73.Google Scholar
- M. Labrèche, De l’approximation harmonique uniforme, Doctoral thesis, Université de Montréal, 1982.Google Scholar
- R. Nevanlinna, Über eine Erweiterung des Poissonschen Integrals, Ann. Acad. Sci. Fenn. Ser.A 24 (4) (1925), 1–15.Google Scholar
- A. Stray, Simultaneous approximation in the Dirichlet space, to appear in Math. Scand. Google Scholar